Question

Show that A x B does not equal B x A, when A and B are nonempty unless A=B.

Answer 1

We will assume A×B = B×A and show that A and B are necessarily the same.

Assume A×B = B×A. Let a ∈ A and b ∈ B. Then (a,b) ∈ A×B. Since A×B = B×A, we have (a,b) ∈ B×A. That is, a ∈ B and b ∈ A. Therefore, a ∈ A implies a ∈ B and b ∈ B implies b ∈ A. So A = B, as we had set out to do.